Paper
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Off-shell structure of twisted (2,0) theory
Authors
Ulf Gran,Hampus Linander
Bengt E. W. Nilsson,Daniel O'Brien,Thomas Flesch,Stephen Neate,Andrzej Z Gorski,Irina Novikova,Jerome Paillet,Theodora Ioannidou,Nerea Zabala Unzalu,Min-Chun Ku,Jacob VanderPlas,Петр Мангилев,Jeffrey Harvey,Joachim Dengler,Anna Paszkowska,Martin Goodchild,Trevor Glasbey,Srinivasan Jagannathan,Saša Obradović,Raul Rosenvald,Michele Portolan,Garrett Poe,Josef Kraus,Kasper Peeters,Alain Jaccard,Carolyn Dzierba,Carolina Abadía Quintero,Marc Torrens,Mikołaj Baranowski,Benjamin Bellenie,Jacek Domagała,Boban Dašić,Andy Van Brocklin,Luis Garcés-Erice,Bernard Gagnon,David Van Wie,Gianluca Calcagni,Vishnu Ram OV,Felipe Lumbreras,N/A ,Aaron Balog,Silviu Pufu,Lorenzo Tancredi,Thorsten Alexander Kern,Christopher Herzog,Martin Köhne,David Clarke,Anselmo Meregaglia,Andre Lukas,Dan Andersen,Fabio Maltoni,Gaurav Sharma,Antonio Ortiz,kanaka b,Igor García Irastorza,Patricia Cleave,Lee Fitzgerald,Hüseyin Bahtiyar,Santiago Folgueras,Polly Livermore,Piret Lõhmus,Maria Jesus Cocero,Phillip Szepietowski,Dean Coble,Sakura Schafer-Nameki,Michele Cicoli,Irene Valenzuela,Ingo Jordan,Ivonne Zavala,Nilanjan Brahma,Radu Popescu,Noah Goldberg,Shlomo Sergei Razamat,Mathew Bullimore,Gaurav Rattan,B. 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Ignatov,Valerie Domcke,Georgios Papathanasiou,Javier Tarrio,pedro candeias,Gerhard Kresbach,Shehu AbdusSalam,Haitang Yang,Michael Katharakis,Laura Andrianopoli,Ryo Saito,Joseph Conlon,Hanna Pawlowska,Antonio Amariti,Alejandro Ibarra,JOSE FRANCISCO DE ASSIS DIAS,Alessandro Minotti,Francisco A. Brito,Zahid Zakir,Gonzalo Torroba,Greg Watchmaker,Mohsen Alishahiha,Ольга Крайнова,Melinda Cosgrove,Maria Carmen Bañuls,Joanna Dziadkowiec,Suresh Govindarajan,Brian Henen,Luigi Cristofolini,Kirk McKenzie,David Tong,Julien Larena,Zhigang Wu,steen hannestad,Enrico Morgante,Victor Rivelles,Thomas Tram,Gabriele Franciolini,Kevin MacDonald,Jessie Shelton,Dariusz Socha,Sinéad O'Connor,Xiang Gao,Wim Hugo,Soo-Jong Rey,Albert Komarov,Kiriaki Chrissopoulou,Stefan Hohenegger,Juan-Ignacio Pozo,Laura Papaleo,Veit Probst,Michelle Fatima Lozada,Lukas Hollenstein,mariemi alonso,Rak-Kyeong Seong,Marcus Tornsö,Li-Mei Liu Liu,Prabhdeep Kaur,Hirofumi Ohyama,Otávio Maia,Kenneth Park,Małgorzata Bartnicka,Borun Chowdhury,Pascal Anastasopoulos,Trent McDonald,Sophie de Buyl,Prarit Agarwal,Partha Mukhopadhyay,Vladimir Mitev,Angel Uranga,Janet Intrieri,Piotr Przybysz,Michael Schmidt,Yasuaki Hikida,alessandro zampini,Diego Zamboni,Iñaki García Etxebarria,Elisa Ruiz Chóliz,Emiliano Molinaro,Hirohiko Masunaga,Thierry PRADIER,Marko Simonovic,Ramesh Natarajan,Nicola Tamanini,Jingwu Duan,Louis Bernier,Marcin Schmidt,Tobias Zingg,Benjamen Kennedy,Gabriele Tartaglino-Mazzucchelli,Shalender Singh,Sandeep Ghosh,Fabian Ruehle,Gim Seng Ng,Chee Sheng Fong,Caio Alves Garcia Prado,Martin Schulc,Amparo Navarro-Faure,Vladimir Kovalenko,Satyajit Jena,Daniel Tapia Takaki,Alessandro Grelli,Richard Barbieri,Renu Bala,Cristina Terrevoli,Nikolay Zaviyalov,Giacomo Volpe,Sabyasachi Siddhanta,Francesca Bellini,Podist Kurashvili,Adela Kravcakova,Sara Vallero,Salvatore Aiola,Jeson Jacob,Valery Kondratiev,Stefano Piano,Roberto Preghenella,Janusz Oleniacz,Arild Velure,Peter Hristov,Jihyun Bhom,Annalisa Guerri,Diane Rigassio Radler,Marek Gutowski,Noriko Ishizaki,Fumichika Uno,Kentaroh Yoshida,Vijay Prabha,Jacob Bourjaily,Tomasz Górecki,Fumihiko Sugino,Marcin Piątek,Anna Horodecka,Benjamin Fuks,Piotr Sołtan,Nikolay Pokrovskiy,Vasiliy Klimenov,David A. 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arXiv:1406.4499v1 [hep-th] 17 Jun 2014
Off-shell structure of twisted (2,0) theory
Ulf Gran, Hampus Linander, Bengt E.W. Nilsson
Department of Fundamental Physics
Chalmers University of Technology
S-412 96 Göteborg, Sweden
ulf.gran@chalmers.se, linander@chalmers.se, tfebn@chalmers.se
Abstract
A Q-exact off-shell action is constructed for twisted abelian (2,0) theory
on a Lorentzian six-manifold of the form M1,5 = C × M4, where C is a flat
two-manifold and M4 is a general Euclidean four-manifold. The properties
of this formulation, which is obtained by introducing two auxiliary fields,
can be summarised by a commutative diagram where the Lagrangian and
its stress-tensor arise from the Q-variation of two fermionic quantities V and
λµν . This completes and extends the analysis in [1].
1
Off-shell structure of twisted (2,0) theory
Ulf Gran, Hampus Linander, Bengt E.W. Nilsson
Department of Fundamental Physics
Chalmers University of Technology
S-412 96 Göteborg, Sweden
ulf.gran@chalmers.se, linander@chalmers.se, tfebn@chalmers.se
Abstract
A Q-exact off-shell action is constructed for twisted abelian (2,0) theory
on a Lorentzian six-manifold of the form M1,5 = C × M4, where C is a flat
two-manifold and M4 is a general Euclidean four-manifold. The properties
of this formulation, which is obtained by introducing two auxiliary fields,
can be summarised by a commutative diagram where the Lagrangian and
its stress-tensor arise from the Q-variation of two fermionic quantities V and
λµν . This completes and extends the analysis in [1].
1
Contents
1 Introduction 2
2 The twisted theory 3
3 Conserved stress-tensor 5
4 Q-exact action 7
4.1 E-sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 F -sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Conclusions 12
A Metric variation of self-dual forms 13
1 Introduction
In this note we consider twisted (2,0) theory on a Lorentzian six-manifold of the
form M1,5 = C × M4 where C is a flat, Lorentzian two-manifold [1]1. This setup
is interesting since it could give some insight into the conjectured correspondence
between four-dimensional gauge theory and two-dimensional CFT known as the
AGT-correspondence [2, 3]. This is one part of a larger web [4–8] of dualities and
relations that can be derived assuming the existence of the elusive superconformal
theory in six dimensions known as (2,0) theory [9, 10].
In previous work [1] the twisted theory on M4 was calculated explicitly in terms
of the free tensor multiplet. It was shown that on a flat background there is a Q-
exact and conserved stress tensor but that these properties did not immediately
extend to a general curved M4. The problem was located to the stress tensor for
the bosonic self-dual two-form which turned out not to be conserved on a general
four-manifold. However, this issue needs to be remedied since the procedure of
topological twisting should result in theories with Q-exact stress tensors defined
on a generally curved background [11–13].
Here we construct an action for the full theory that is Q-exact off-shell using
two different kinds of auxiliary field. The free theory splits into two parts, one
of which is equivalent to Donaldsson-Witten theory, and hence this sector can be
taken off-shell following the standard techniques described in [14–16].
1The twisting is carried out in Minkowski signature where the non-compact part of the Lorentz
group prevents a full twist but where a low energy limit still produces supercharges with the
required properties for a topological theory.
2
1 Introduction 2
2 The twisted theory 3
3 Conserved stress-tensor 5
4 Q-exact action 7
4.1 E-sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 F -sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Conclusions 12
A Metric variation of self-dual forms 13
1 Introduction
In this note we consider twisted (2,0) theory on a Lorentzian six-manifold of the
form M1,5 = C × M4 where C is a flat, Lorentzian two-manifold [1]1. This setup
is interesting since it could give some insight into the conjectured correspondence
between four-dimensional gauge theory and two-dimensional CFT known as the
AGT-correspondence [2, 3]. This is one part of a larger web [4–8] of dualities and
relations that can be derived assuming the existence of the elusive superconformal
theory in six dimensions known as (2,0) theory [9, 10].
In previous work [1] the twisted theory on M4 was calculated explicitly in terms
of the free tensor multiplet. It was shown that on a flat background there is a Q-
exact and conserved stress tensor but that these properties did not immediately
extend to a general curved M4. The problem was located to the stress tensor for
the bosonic self-dual two-form which turned out not to be conserved on a general
four-manifold. However, this issue needs to be remedied since the procedure of
topological twisting should result in theories with Q-exact stress tensors defined
on a generally curved background [11–13].
Here we construct an action for the full theory that is Q-exact off-shell using
two different kinds of auxiliary field. The free theory splits into two parts, one
of which is equivalent to Donaldsson-Witten theory, and hence this sector can be
taken off-shell following the standard techniques described in [14–16].
1The twisting is carried out in Minkowski signature where the non-compact part of the Lorentz
group prevents a full twist but where a low energy limit still produces supercharges with the
required properties for a topological theory.
2
In the other sector there is a self-dual tensor field whose presence in the Q
transformation rules leads to an unwanted metric dependence. However, by the
introduction of an auxiliary vector field we are able to eliminate this metric de-
pendence in a similar fashion as in [17]. Also in this sector, this step leads to a
formulation where the scalar supercharge is nilpotent off-shell and after construct-
ing a Q-closed and covariant Lagrangian for the entire theory these properties
become manifest also for the stress-tensor. For the bosonic self-dual two-form we
have also added certain curvature terms to its equation of motion, which becomes
possible after the twisting is performed. Note that these terms cannot be obtained
in the original (2,0) theory in six dimensions but they are known from the closely
related interacting theory constructed in five dimensions in [5].
With these modifications of the theory that is naively obtained from twisting
the six-dimensional (2,0) theory on M4, we find an off-shell theory whose metric
variations and Q transformations commute. This feature then implies that the
stress tensor can be derived from a fermionic quantity V (given below) either by
going via the Lagrangian or via λµν (where T µν = {Q,λµν }). In section 4 this
is summarized in a commuting square whose corners represent the four quantities
involved, i.e., T µν , λµν , V and its Q transform, the Lagrangian.
In section 2 we review the four-dimensional theory obtained by twisting the
six-dimensional (2,0) theory on C × M4. The problem encountered previously and
its resolution are briefly explained in section 3. In section 4 we construct an off-
shell formulation including a Q-exact action. Finally, in section 5 we summarise
and comment on the results.
2 The twisted theory
For the convenience of the reader we here give a short review of the twisted the-
ory, for details see [1]. On a general background the six-dimensional (2,0) theory
admits no twist that preserves any supersymmetry since the Spin(5) R-symmetry
cannot be used to fully twist the supercharges transforming in the larger six di-
mensional Lorentz group Spin(1,5). However, on specific backgrounds such as the
one considered here of the form M6 = C × M4, the Lorentz group is small enough.
Here we twist by considering a new SU(2)′ as the diagonal embedding
SU(2)′ = SU(2)r × SU(2)R , (1)
where the six dimensional Lorentz group is Spin(1,1) × SU(2)l × SU(2)r and the
R-symmetry subgroup is given by
SU(2)R × U(1)R ∼= Spin(3) × Spin(2) ⊂ Spin(5)R. (2)
3
transformation rules leads to an unwanted metric dependence. However, by the
introduction of an auxiliary vector field we are able to eliminate this metric de-
pendence in a similar fashion as in [17]. Also in this sector, this step leads to a
formulation where the scalar supercharge is nilpotent off-shell and after construct-
ing a Q-closed and covariant Lagrangian for the entire theory these properties
become manifest also for the stress-tensor. For the bosonic self-dual two-form we
have also added certain curvature terms to its equation of motion, which becomes
possible after the twisting is performed. Note that these terms cannot be obtained
in the original (2,0) theory in six dimensions but they are known from the closely
related interacting theory constructed in five dimensions in [5].
With these modifications of the theory that is naively obtained from twisting
the six-dimensional (2,0) theory on M4, we find an off-shell theory whose metric
variations and Q transformations commute. This feature then implies that the
stress tensor can be derived from a fermionic quantity V (given below) either by
going via the Lagrangian or via λµν (where T µν = {Q,λµν }). In section 4 this
is summarized in a commuting square whose corners represent the four quantities
involved, i.e., T µν , λµν , V and its Q transform, the Lagrangian.
In section 2 we review the four-dimensional theory obtained by twisting the
six-dimensional (2,0) theory on C × M4. The problem encountered previously and
its resolution are briefly explained in section 3. In section 4 we construct an off-
shell formulation including a Q-exact action. Finally, in section 5 we summarise
and comment on the results.
2 The twisted theory
For the convenience of the reader we here give a short review of the twisted the-
ory, for details see [1]. On a general background the six-dimensional (2,0) theory
admits no twist that preserves any supersymmetry since the Spin(5) R-symmetry
cannot be used to fully twist the supercharges transforming in the larger six di-
mensional Lorentz group Spin(1,5). However, on specific backgrounds such as the
one considered here of the form M6 = C × M4, the Lorentz group is small enough.
Here we twist by considering a new SU(2)′ as the diagonal embedding
SU(2)′ = SU(2)r × SU(2)R , (1)
where the six dimensional Lorentz group is Spin(1,1) × SU(2)l × SU(2)r and the
R-symmetry subgroup is given by
SU(2)R × U(1)R ∼= Spin(3) × Spin(2) ⊂ Spin(5)R. (2)
3
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